Prandtl Meyer Shock problem
Objective
- To visualize the Prandtl Meyer expansion fan over an expansion corner using Converge and Paraview software and also understand the effect of Sub-Grid Scale(SGS) on the shock location.
- To study types of boundary conditons used for shock problems.
- To understand the reason of usage of Neumann boundary condition at the outlet.
- To understand what are shock waves and the physics behind them.
- To understand the effect of using of subsonic initial condtion for supersonic flow.
Shock wave
When the speed of a source exceeds the speed of sound, the wave fronts lag behind the source in a cone-shaped region with the source at the vertex. The edge of the cone forms a supersonic wave front with an unusually large amplitude called a \"shock wave\".
Unlike ordinary sound waves, the speed of a shock wave varies with its amplitude. The speed of a shock wave is always greater than the speed of sound in the fluid and decreases as the amplitude of the wave decreases. When the shock wave speed equals the normal speed, the shock wave dies and is reduced to an ordinary sound wave.
Speeds less than the speed of sound have a Mach numbers between zero and one and are described as subsonic. Those greater than the speed of sound have Mach numbers greater than one are described as supersonic. Speeds approximately equal to the speed of sound have Mach numbers approximately equal to one and are described as transonic.In supersonic waves, there is a special class of waves called hypersonic waves. They travel with speeds for which Mach number is greater than or equal to 5.
Types of shock waves
- Normal shock - Shock created at 90 degrees.
- Oblique shock - Shock created at an angle as per body\'s deflection angle.
- Bow shock - Shock created when flow velocity exceeds Mach number=1.
Physics behind shock waves
The ratio of the speed of a moving object (v) to the speed of sound (c) in a fluid is known as the Mach Number (Ma), i.e, Ma=(v/c).
The propagation of disturbances in subsonic and supersonic flows: V is the velocity of flow, a is the acoustic speed at local temperature.
(a)The flow in subsonic speeds has prior knowledge of the disturbance, the streamlines diverge and go around the body.
(b)However in supersonic flow because of the presence of shock waves ahead of the body, the velocity of flow suddenly reduces to subsonic values after the shock wave and this informs the fluid streamlines about the presence of the body. In supersonic flow, shock waves perform the dual function of information transfer as well as a mechanism by which the flow can turn towards itself.
Types of boundary condtions used
Three types of boundary conditions are used to solve shock wave problems:
- Dirichlet boundary condition: It is a type of boundary condition which when imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take along the boundary domain.
- Neumann boundary condition: It is a type of boundary condition which when imposed on an ordinary or a partial differential equation, the condition specifies the value in which the derivative of a solution is applied within the boundary of the domain.
As all the characteristic lines travel out of the domain from the outlet boundary, the only way of defining the condition at the outlet boundary is through Neumann boundary condition.
- Mixed boundary condition: For a PDE this defines a boundary value problem in which the solution of the given equation is required to satisfy different boundary condition on disjoint parts of the boundary of the domain where the condition is stated.
Prandtl Meyer expansion fan
The geometry is checked for all the errors by running diagnosis and then its flagged into different boundaries to further setup the case to run simulation.
Boundary Flagging
Case setup
- It is a time-based case.
- Air is the pre-defined mixture used here.
- Simulation run parameters are steady-state solver, full hydrodynamic and compressible gas flow solver.
- In steady-state monitor the variable used for monitoring is total pressure which is monitored at outlet.
- Simulation time parameters:
start time : 0 cycles
end time : 25000 cycles
Initial time-step : 1e-7 sec
Minimun time-step : 1e-7 sec
Maximum time-step : 1 sec
- Boundary Condiitons :
Inlet boundary:It is assigned inflow boundary condition with pressure set to Zero Normal Gradient(NE) and temperature boundary condition set to 286.1K. Velocity boundary condition is set to 680m/s along x-direction.
Outlet boundary:It is assigned outflow boundary condition with pressure and velocity boundary condition set to Zero Normal Gradient(NE).
Top and bottom boundaries :It is assigned with wall conditions with slip condition.
Front and back boundaries :These are 2D boundaries. They are alloted 2D boundary condition as they are in the z-direction and along z-direction simulation won\'t be considered as we perform 2D simulation.
- Region and Initialization : Region 0 is the volumetric flow region assigned to all the boundaries. Only single boundary will be used in this case. Here velocity is set to:
Case 1: 100m/s along x-direction for subsonic conditon.
Case 2: 680m/s along x-direction for supersonic conditon.
- Turbulence modelling : RNG k-ɛ
- Grid control: Grid size - dx = dy = dz =0.8m.
- Adaptive Mesh Refinement (AMR):
Case 1: For subsonic flow, SGS is set 0.05K.
Case 2: For supersonic flow, SGS is set to 0.05K, 0.1K and 0.2K.
Adaptive Mesh Refinement.
Adaptive Mesh Refinement is used to refine the mesh locally in the region where shock would occur. In other regions the mesh size remains the same as the base grid size defined by the user but for the regions where shocks would occur finer cells would be required.
Here temperature based mesh refinement of 2 embed layers is used that would monitor temperature curvature. When in a region the curvature is greater than the value defined by the user then the mesh would be refined in that region. For instance, for SGS value of 0.05K, when in a region SGS value would be greater than 0.05K then in that region the cells would be refined maximum upto 2 embedding layers.
Since the embedding level used here is 2(n), then the smallest cell size that would be used in the mesh refinement will be,
Grid size = 0.8m(Mesh size) / 2^2(n) = 0.2m.
Grid size of 0.2m would be the smallest grid size to be used for mesh refinement.
- Output Files: 3D data files were dumped every 100 cycles.
Case 1: Subsonic
Meshing with Adaptive Mesh Refinement (AMR)
Results
Cell count
Rank 0, rank 1 , rank 2 and rank 3 are the count of cells for four processors.
Pressure
Total pressure at inlet and outlet.
Mach number
Mach number at inlet and outlet.
Velocity
Velocity at inlet and outlet.
Velocity contour plot without mesh.
Velocity contour plot with mesh.
Temperature
Temperature at inlet and outlet.
Temperature contour plot without mesh.
Temperature contour plot with mesh.
Case 2: Supersonic
- SGS = 0.05K
Meshing with Adaptive Mesh Refinement (AMR)
Results
Cell Count
Rank 0, rank 1 , rank 2 and rank 3 are the count of cells for four processors.
Pressure
Total pressure at inlet and outlet.
Velocity
Velocity at inlet and outlet.
Velocity contour plot without mesh.
Velocity contour plot with mesh.
Mach Number
Mach number at inlet and outlet.
Temperature
Temperature at inlet and outlet.
Temperature contour plot without mesh.
Temperature contour plot with mesh.
- SGS = 0.1K
Meshing with Adaptive Mesh Refinement (AMR)
Results
Cell Count
Rank 0, rank 1 , rank 2 and rank 3 are the count of cells for four processors.
Pressure
Total pressure at inlet and outlet.
Mach number
Mach number at inlet and outlet.
Velocity
Velocity at inlet and outlet.
Velocity contour plot without mesh.
Velocity contour plot with mesh.
Temperature
Temperature at the inlet and outlet.
Temperature contour plot without mesh.
Temperature contour plot with mesh.
- SGS = 0.2K
Meshing with Adaptive Mesh Refinement (AMR)
Results
Cell count
Rank 0, rank 1 , rank 2 and rank 3 are the cell counts for four processors.
Pressure
Pressure at inlet and outlet.
Mach number
Mach number at inlet and outlet.
Velocity
Velocity at inlet and outlet.
Velocity contour plot without mesh.
Velocity contour plot with mesh.
Temperature
Temperature at inlet and outlet.
Temperature contour plot without mesh.
Temperature contour plot with mesh.
Mesh refinement doesn\'t start immediatly, after running simualtion for few cycles the mesh gets refined according to the SGS value given. This is because in the initial conditions of AMR it is provided that the mesh gets refined after 5000 cycles.
Velocity and Temperature contour plots animation for Supersonic flows
- SGS = 0.05K
- SGS = 0.1K
- SGS = 0.2K
Conclusion
Subsonic Flow (Case 1)
- Using subsonic initial condition at a velocity of 100m/s for supersonic flow results into the blowing up of the simulation (around 12604 cycle) due to the huge difference between the inlet velocity and initial condition.
- If one uses the subsonic initial condition for the supersonic flow then huge difference between the inlet velocity (680m/s) and initial condition (100m/s) lead the solution into the wrong direction and the solution diverges.
Supersonic Flow (Case 2)
- In all the three flow cases the Mach number increases as the flow passes through the expansion fan whereas other properties like temperature, pressure, and velocity tends to decrease.
- It is also observed that for SGS value of 0.2K the shock waves are not well refined as there is almost no mesh refinement along the shock path. Whereas in case of SGS value of 0.05K the shock waves are highly refined as in this case along the shock path mesh is very fine.
- From post processing all the cases it can be observed that with the increasing value of SGS the expansion fan tends to become broad and less distinct. The reason behind this is that the adaptive mesh refinement (AMR) is dependent on the value of SGS. With the higher value of SGS the mesh refinement required to capture the expansion waves gets limited which results into the blurry and not well-defined waves.
- With the decreasing SGS number the count for total cells keeps on increasing.This is because the lower value of SGS ensures higher mesh refinement which in turn increases the cell count.